Calculus requires that one treat the infinite as
though it were a definite quantity. The infinite,
however, is not a definite quantity; it denotes a
process that can be continued without end, as Zeno
demonstrated with his famous paradoxes. This
fundamenal misunderstanding of the nature of the
infinite leads calculus to speak of such nonsense as
"speed at an instant" (not to mention, "light is a
wave and a particle"). Because of this fundamental
error regarding the infinite, calculus can NEVER
provide an intelligible account of the universe. At
best, it is a very useful approximation. At worst, it
is simply stupid.
Discuss.

"Calculus requires that one treat the infinite as though it were a definite quantity."
No. You can base calculus on a system ofnumbers with infinitely small and big quantitites, the hyperreal number system, but that's not the most common approach. Usually, calculus is about finite numbers.
"The infinite, however, is not a definite quantity; it denotes a process that can be continued without end,"
That's why you talk about limits all the time in calculus.
"as Zeno demonstrated with his famous paradoxes."
Actually these paradoxes show little but how confused precalculus notion of movement are.
"This fundamenal misunderstanding of the nature of the infinite leads calculus to speak of such nonsense as "speed at an instant" (not to mention, "light is a wave and a particle")."
1. Calculus is abot real numbers, not about physics. 2. Speed at an instant means nothing but that at an certain instant, speed, which changes all the time, can be approximated arbitrarily good.

"Actually these paradoxes show little but how confused precalculus notion of movement are."
I *strongly* disagree. Don't the paradoxes show that one cannot think of the infinite as a definite quantity and that time and space are continuous, not made up of an "infinite number" of "infinitely small" parts? The use of modern calculus has obscured this point.
You are right that I am confounding "calculus" with "physicists who use calculus". This was my mistake, and I apologize. I have no objection to calculus on its own. But on its own, it says nothing about nature. I object to people, like physicists, who try to make statements about reality based on calculus.
As you rightly point out, calculus speaks of "limits". A limit is that point that can NEVER be reached, right? So, limits do not actually exist in nature, right? Nothing is ever "at a limit". But physicists often speak of these limits as though they do exsits, when they say such things as "speed at an instant". The instant is a limit, and thus it does not actually exist. Furthermore, time is not made up of instants, and it is most certainly nonsense to speak of a "speed at an instant".
"Speed at an instant means nothing but that at an certain instant, speed, which changes all the time, can be approximated arbitrarily good."
As I said, the instant is a limit, so it doesn't actually exist. It is that point that is NEVER reached. Otherwise, it sounds like you are agreeing with me, when I said that calculus is "At best... a very useful approximation."

"Don't the paradoxes show that one cannot think of the infinite as a definite quantity and that time and space are continuous, not made up of an "infinite number" of "infinitely small" parts? The use of modern calculus has obscured this point."
The point of using calculus is to model the physical world as having continous time. The "paradoxes" are only paradoxical because it wasn't known how to do the math. One can prove that it's contradictory to assume that Archilles will never pass that turtle.
"As you rightly point out, calculus speaks of "limits". A limit is that point that can NEVER be reached, right?"
No. A limit in mathematics is a purely abstract term. What does "reach" mean? Here is a definition of the limit of a fuction in calculus. One can define limits in more general settings, but that's of little philosophical interest.
"So, limits do not actually exist in nature, right?"
I agree, but I don't think any numbers exist like, say, cats and dogs exist. Numbers are part of our language for talking about the world, they aren't the world.
"But physicists often speak of these limits as though they do exsits, when they say such things as "speed at an instant". The instant is a limit, and thus it does not actually exist."
What they mean when they say the speed at the instant t is v is that if one would measure the position of an object befor and after time t, the resulting speed is close to v. The shorter the span used for measuring, the closer the measured speed is to v.
"calculus is "At best... a very useful approximation.""
Calculus is based on the idea of approximation. But that's no reason to ban terms like "speed at an instant".
Furthermore, time is not made up of instants, and it is most certainly nonsense to speak of a "speed at an instant".

"The "paradoxes" are only paradoxical because it
wasn't known how to do the math. One can prove that
it's contradictory to assume that Archilles will never
pass that turtle."
Again, I think you have seriously misunderstood the
paradoxes. Zeno does NOT "assume that Achilles will
never pass the turtle". He starts by assuming that he
can have divided space infinitely (i.e. that infinity
is a definite quantity; that it can be actualized) and
then he shows what paradoxes result. The paradoxes
are reductio ad absurdums that show that infinity is
not a definite quantity and that space and time are
continuous, not discrete. Furthermore, Aristotle
(circa 4th century BC, just a wee bit prior to the
invention of calculus) explained this about Zeno's
paradoxes. So, you are clearly mistaken when you say
that the paradoxes are paradoxical "because it wasn't
known how to do the math", unless you are suggesting
that Aristotle had some secret knowledge of calculus.

"No. A limit in mathematics is a purely abstract term.
What does "reach" mean? Here is a definition of the
limit of a fuction in calculus. One can define limits
in more general settings, but that's of little
philosophical interest."
I do not care for you definition. It lacks words :)
I was refering to the definition of a limit as
expressed by Newton, who (I believe) invented calculus
and was kind enough to express his definitions simple
words. With respect to velocity and time, I believe
he said that, as you decrease the time, the limit is
that speed which your average speed continually
approaches, and which you can always get closer to,
but which you can NEVER attain, or "reach" (because
there is no "average speed" at an instant). I think
that if you put those mathamtical symbols into words,
you will get a similar defintion (although more
general).

"Again, I think you have seriously misunderstood the paradoxes. Zeno does NOT "assume that Achilles will never pass the turtle". He starts by assuming that he can have divided space infinitely (i.e. that infinity is a definite quantity; that it can be actualized) and then he shows what paradoxes result."
But no paradox results. You can model time as the points on the real line and still see Archilles pass that turtle.
"The paradoxes are reductio ad absurdums that show that infinity is not a definite quantity and that space and time are continuous, not discrete."
An continuous means what in this context?
"Furthermore, Aristotle (circa 4th century BC, just a wee bit prior to the invention of calculus) explained this about Zeno's paradoxes. So, you are clearly mistaken when you say that the paradoxes are paradoxical "because it wasn't known how to do the math", unless you are suggesting that Aristotle had some secret knowledge of calculus."
Exactly. Now with calculus we see that there is no paradox watsoever.

"I agree, but I don't think any numbers exist like,
say, cats and dogs exist. Numbers are part of our
language for talking about the world, they aren't the
world."
This is interesting... how could cats possibly exist
without numbers (note the plurality of catS!). How
could you have "many" of anything with numbers?
Do at least believe that the One and the Many exist in
the the world? If not, I think you'll have a hard
time finding even ONE cat. Without the One and the
Many, there is no world. All becomes unspeakable
flux!

"I was refering to the definition of a limit as expressed by Newton, who (I believe) invented calculus and was kind enough to express his definitions simple words."
The development of calculus didn't end with Newton. The mathematics of Newton weren't rigorous by modern standards.
"With respect to velocity and time, I believe he said that, as you decrease the time, the limit is that speed which your average speed continually approaches, and which you can always get closer to, but which you can NEVER attain, or "reach" (because there is no "average speed" at an instant)."
I don't see the role of the "never attain" in the definition. What does that mean? The notion of an actual speed at a time is consistent but unnecessary.

"You can model time as the points on the real line and
still see Archilles pass that turtle."
How many points are on a one inch line?
"An continuous means what in this context?"
Not composed of discrete parts.
"Exactly. Now with calculus we see that there is no
paradox watsoever."
What? I just said that Aristotle saw this 2000 years
before calculus was invented. I don't understand what
you are agreeing with.

"The development of calculus didn't end with Newton.
The mathematics of Newton weren't rigorous by modern
standards."
That's a pretty vague criticism (which I am not sure
is even fair). Regardless, has the idea of a limit
changed since he invented it? I doubt it has, but if
you claim it has, how so?
"I don't see the role of the "never attain" in the
definition. What does that mean? The notion of an
actual speed at a time is consistent but unnecessary."
I think it means that the limit is that which is
continually approached by a process. If this is still
confusing, perhaps it would be clarified if you
described to me, in words, what you are actually doing
when you determine speed at an instant.

"This is interesting... how could cats possibly exist without numbers (note the plurality of catS!). How could you have "many" of anything with numbers? Do at least believe that the One and the Many exist in the the world? If not, I think you'll have a hard time finding even ONE cat. Without the One and the Many, there is no world. All becomes unspeakable flux!"
'Fritz is a cat and Max is a cat. Fritz is not Max.' I've just stated the plurality of cats without mentioning numbers ore "one and many". It's that easy. No hard time.
"How many points are on a one inch line?"
You confuse mathematics and the world. Mathematics is part of the language.

"That's a pretty vague criticism (which I am not sure is even fair). Regardless, has the idea of a limit changed since he invented it? I doubt it has, but if you claim it has, how so?"
It has been clarified. Newton didn't use them very explicitely in his proofs.
""I don't see the role of the "never attain" in the definition. What does that mean? The notion of an actual speed at a time is consistent but unnecessary."
I think it means that the limit is that which is continually approached by a process.""
That's not what I wonderr about. Where does the "never reach" part come i?
"If this is still confusing, perhaps it would be clarified if you described to me, in words, what you are actually doing when you determine speed at an instant."
I cannot determine the speed at an instant. It's a if then statement. If I measure the average speed in a time interval short enough, I get as close to the speed at an instant as I want.

"'Fritz is a cat and Max is a cat. Fritz is not Max.'
I've just stated the plurality of cats without
mentioning numbers ore "one and many". It's that easy.
No hard time."
In saying that Fritz is "a cat", you have taken many
different things (the ears of Fritz, the eyes of
Fritz, the tail of Fritz, etc) and UNIFIED them, that
is you have declared that they all are ONE thing,
namely a cat. But to declare that something is a
unity is to say nothing else than it is ONE. Just
because you said "a" cat (as opposed to "one" cat)
doesn't change the fact that the One is necessary for
your statement, as is the Many. The fact that you
didn't explicitly say "one" is irrelevant.

"In saying that Fritz is "a cat", you have taken many different things (the ears of Fritz, the eyes of Fritz, the tail of Fritz, etc) and UNIFIED them, that is you have declared that they all are ONE thing, namely a cat."
And if I say "X is a stone" I have somehow unified the molecules?
I made no ontological commitment as to what a cat is. All I need is a equality relation. This again is part of language not the world. So why does "the one and the many " exist in the world?

"It has been clarified. Newton didn't use them very
explicitely in his proofs."
Again, this is a suspicously general criticism, which
seems false to me. Newton has a three page Scholium
that explains what a limit is. Three pages... that's
pretty explicit. But, if you persist in disagreeing,
perhaps you could say what he left out of his three
page definition, that has been clarified by modern
mathematicians.
"I cannot determine the speed at an instant. It's a if
then statement. If I measure the average speed in a
time interval short enough, I get as close to the
speed at an instant as I want."
How could possibly know that you are getting closer to
the speed at an instant, if you "cannot determine the
speed at an instant"?
Furthermore, you say, "I get as close to the speed at
an instant as I want". What I am saying is that,
although you can get closer, you cannot reach it. You
must know what I mean by reach it, if you yourself
speak of getting closer to it, right?

"Again, this is a suspicously general criticism, which seems false to me. Newton has a three page Scholium that explains what a limit is. Three pages... that's pretty explicit. But, if you persist in disagreeing, perhaps you could say what he left out of his three page definition, that has been clarified by modern mathematicians."
He didn't use them in his proofs. Please read more carefully.
"How could possibly know that you are getting closer to the speed at an instant, if you "cannot determine the speed at an instant"?"
I cannot. It's a scientificial theory that can be tested bbut not proven with a finite data set.
"Furthermore, you say, "I get as close to the speed at an instant as I want". What I am saying is that, although you can get closer, you cannot reach it. You must know what I mean by reach it, if you yourself speak of getting closer to it, right?"
No. If the speed remains constant than I can indeed reach the speed at an instant.

"And if I say "X is a stone" I have somehow unified
the molecules?"
Yes.
"I made no ontological commitment as to what a cat is.
All I need is a equality relation. This again is part
of language not the world. So why does "the one and
the many " exist in the world?"
You need not make any ontological commitment to what a
cat is. All you need to say is that Fritz is a thing;
he is a whole; he is unified. By saying this, you
grant that he is one, such as to distinguish him from
MANY other ONES, like Max.
"He didn't use them in his proofs. Please read more
carefully."
Read more carefully? Fair enough. I will. Now that I
have, it is clear to me that you are simply wrong. He
DID use limits in his proofs. Repeatedly. Look at
Principia, Book 1, Proposition 1 (this is directly
following his three page scholium explaining what he
means by "limit") He uses it in the very first
proposition! Have you read the Principia?
"No. If the speed remains constant than I can indeed
reach the speed at an instant."
To what does your "No" refer? I assume you now know
what I mean by reach, since you used the term
yourself. But how can you reach speed at an instant,
even with constant speed? There is no speed at an
instant! What speed could a thing possibly have at an
instant, 5 miles per NOTHING?

"By saying this, you grant that he is one, such as to distinguish him from MANY other ONES, like Max."
And how does that show the existence of "the one and the many"? I guess you aren't exactly a nominalist.
"Repeatedly. Look at Principia, Book 1, Proposition 1 (this is directly following his three page scholium explaining what he means by "limit") He uses it in the very first proposition! Have you read the Principia?"
There is no explicit limiting argument in proposition 1. He simply assumes that a limit exist. The part before doesn't really define limits but states some Lemmata. The definition is only implicit.
"But how can you reach speed at an instant, even with constant speed? There is no speed at an instant! What speed could a thing possibly have at an instant, 5 miles per NOTHING?"
You measure the average speed nearer and neare the instant- and reach it since it's constant near that instant.
"What speed could a thing possibly have at an instant, 5 miles per NOTHING?""
It could have, say, 5 miles per hour.

"And how does that show the existence of "the one and
the many"? I guess you aren't exactly a nominalist."
If you say that the cat is, and you thereby grant that
he is one, how can you say that the One is not? How
could the cat be one if the One did not exist? You
would be saying that the cat is one, but the one is
not, and thus the cat is that which is not. The cat
would both be one and not be one; You would be denying
the law of non-contradiction, which is a no-no.
BTW, I had to look up "nominalist". It seems I am
just about the opposite.
"There is no explicit limiting argument in proposition
1. He simply assumes that a limit exist. The part
before doesn't really define limits but states some
Lemmata. The definition is only implicit."
After Lemma 11, there is a Scholium where he explains
at length what he means by "limit". This explanation
is not implicit, but explicit and lengthy. Moreover,
you said that "he didn't use limits in his proofs".
But Prop 1 clearly rests on his use of limits. Look
at the last step. The whole Propisition rests on his
use of limits.
"You measure the average speed nearer and neare the
instant- and reach it since it's constant near that
instant."
You see, calculus has confused you. Even if I grant
that there is such a thing as an "instant", there
could be NO motion in this "instant", because motion
requires the elapsing of time. But there is no
elapsing of time in an instant, because all elapsed
time can be divided into smaller parts, but an instant
is indivisible. So, there can clearly be no motion in
an instant. But without any motion, there can be
speed, for speed is a measure of motion. So, there
clearly can be no "speed at an instant". It is simply
nonsense.
That is why a limit must be understood as that which
one can never reach, because once you speak of
reaching it, you start speaking nonsense. And that is
why calculus cannot be used to accurately describe
reality, because it relies on limits which do not
exist in reality, and thereby perverts the nature of
infinitey, time, and space.

"BTW, I had to look up "nominalist". It seems I am just about the opposite."
I know. But that would be another discussion.
"You see, calculus has confused you. Even if I grant that there is such a thing as an "instant", there could be NO motion in this "instant", because motion requires the elapsing of time."
And? That instant doesn't stand alone.
"But there is no elapsing of time in an instant, because all elapsed time can be divided into smaller parts, but an instant is indivisible."
"Speed at an instant" is about the local behavior of speed around an instant. Of course you cannot separate the point from the points lying near. But nobody does this.
"That is why a limit must be understood as that which one can never reach, because once you speak of reaching it, you start speaking nonsense."
You reach the speed, not the instant. You confuse two different things here.
"And that is why calculus cannot be used to accurately describe reality, because it relies on limits which do not exist in reality, and thereby perverts the nature of infinitey, time, and space."
It is based on smoothness, not on limits. There are other approaches that come to the same results. Calculus is simply a mathematical field. The methods of calculus can be succesfully be used in physics. What more do you want? Given that we cannot measure anything with arbitrary precision all this has no relation to practical science whatsoever.

"And? That instant doesn't stand alone."
If there is no motion in an instant, it doesn't matter how many instants "stand together"; there still will be no motion. Nothing + Nothing = Nothing
""Speed at an instant" is about the local behavior of speed around an instant. Of course you cannot separate the point from the points lying near. But nobody does this."
The limit corresponds to the exact speed at a precise moment in time, the speed at an instant. It is derived by observing the local behavior of speed around an instant, but the limit is the speed at a point, seperated from other points.
"You reach the speed, not the instant. You confuse two different things here."
When do you reach the speed, if not when you reach the instant? Do reach the speed at an instant before you reach the instant? After? They are simultanious events.
"It is based on smoothness, not on limits. There are other approaches that come to the same results. Calculus is simply a mathematical field. The methods of calculus can be succesfully be used in physics. What more do you want?"
The use of limits is essential to calculus. Limits, however, are based on a misunderstanding of the infinite--they treat the infinite as though it is a definite quantity that can be reached. When calculus is applied to physics, it leads people to start speaking of the limits as though they exist in nature; people start speaking of "instants", which are limits i.e. the infinitely small pieces of time. But "infinitely small pience of time" is a nonsensical phrase. Thinking of time in this way leads to all sorts of contradictions. This is just what Zeno was illustrating with his paradoxes. I agree that calculus can be used in physics, but only as an approximation, one that can give us no intelligible account of the universe because it is based on a fundamental misuderstanding of the nature of time and space. Ptolomy was able to predict the motions of the heavens more accurately than Copernicus. That doesn't mean the earth is at the center of the universe and the sun travels around it in a perfect circle.
So, what more do I want? An intelligible account of the universe, not mere predictive power based approximations.

"If there is no motion in an instant, it doesn't matter how many instants "stand together"; there still will be no motion."
It does matter, that´s why differentiable functions are all continuos.
"Nothing + Nothing = Nothing"
This means?
"The limit corresponds to the exact speed at a precise moment in time, the speed at an instant. It is derived by observing the local behavior of speed around an instant, but the limit is the speed at a point, seperated from other points."
No. With different other points, you get different speeds at that instant (or none at all).
"When do you reach the speed, if not when you reach the instant? Do reach the speed at an instant before you reach the instant? After? They are simultanious events."
The speed is a number. The sequence of average speeds around an instant can eventually become constant, hence the speed at an instant gets reached.
"The use of limits is essential to calculus. Limits, however, are based on a misunderstanding of the infinite--they treat the infinite as though it is a definite quantity that can be reached."
No quantity you talk about in ordinary calculus, based on limits, is infinite. Maybe you are thinking of non-standard analysis where infinitesimals and their infinite converses are part of the number system. The point of limits was not to use infinitesimals.
"When calculus is applied to physics, it leads people to start speaking of the limits as though they exist in nature; people start speaking of "instants", which are limits i.e. the infinitely small pieces of time."
Since most people are nominalists now, there should be no confusion.
"But "infinitely small pience of time" is a nonsensical phrase. Thinking of time in this way leads to all sorts of contradictions."
While there is no need for calculus to speak about infinitely small quantities, they lead to no contradictions whatsoever. This follows from the compactness theorem of first order logic.
"This is just what Zeno was illustrating with his paradoxes."
Zeno demonstrated his mathematical inability and little more.
"I agree that calculus can be used in physics, but only as an approximation,"
Since one cannot measure anything with arbitrary precision, one cannot differ between sufficiently good approximations and "reality".
"one that can give us no intelligible account of the universe because it is based on a fundamental misuderstanding of the nature of time and space."
I think the nature of time and space is an issue of physics, not armchair philosophy. How come one cannot derive Zenos paradoxes from the models used in physics?
"Ptolomy was able to predict the motions of the heavens more accurately than Copernicus. That doesn't mean the earth is at the center of the universe and the sun travels around it in a perfect circle."
We can actually measure the difference between the predictions and the observed outcome. This is a different story.
"So, what more do I want? An intelligible account of the universe, not mere predictive power based approximations."
Than I feel sorry for you. You want something you will never have.
So, what more do I want? An intelligible account of the universe, not mere predictive power based approximations.

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